The objective of this project is to show that the methods of X-ray diffraction analysis of biological macromolecules can be automated by methods which place less emphasis on multiple isomorphous crystal structures. We propose to derive and test on solved protein structures, mathematical methods for estimating phases of structure factors which (1) use relationships between isomorphous replacement and the ab initio or direct phasing methods, (2) employ a different theory for describing the nature of possible errors in the isomorphous replacement and anomalous scattering systems, and (3) imbed generalized crystallographic symmetry (true and non-crystallographic) into the formulas. The methods will be derived, translated into an appropriate computer language, and implemented on the computer facilities at the Medical Foundation of Buffalo and the National Resource for Computation in Chemistry. The methods will then be applied to synthetic data obtained from protein coordinates deposited in the Protein Data Bank at the Brookhaven National Laboratory and to the real data sets of bacterial ferrodoxin and 2-zinc insulin. Accuracy of the estimated phases will be assessed by conventional statistical methods and by examination of the computed electron density functions. Except for crystallizing the macromolecule itself, the critical opartional step in the diffraction analysis has been the chemical preparation and analysis of a series of isomorphous derivative crystal structures. The time invested in preparing and analyzing several derivatives constitutes a large fraction of the time spent in a complete structure determination. Since the proposed methods de-emphasize the role of the isomorphous derivative, adequate phase estimates may be obtained from a reduced number of derivatives. This would lead to studies predicated on a knowledge of a series of structures, since less time would be spent on the determination of each one. New areas previously studied mostly from the functional side of the structure-function equation would now be more susceptible to detailed structural investigations.